Page - 93 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 68 6,831 12,206 16,829 13,051 26,102 39,153 0 10,000 20,000 30,000 40,000 50,000 2 4 6 Machines per stage Algorithm Industry Figure25.EopofB&Balgorithmandindustry. Table19.Resultsof the industryandB&Balgorithm. MachinesperStage Industry B&BAlgorithm Eop (kW) Cmax (h) Eop (kW) Cmax (h) 2 13,050.90 186 6,831.40 96.52 4 26,101.80 93 12,206.42 43.82 6 39,152.70 62 16,828.72 25.57 Table20.The industryandbi-objectiveGAdegradationoverB&Balgorithm(%). MachinesperStage Industry Bi-ObjectiveGA %Eop %Cmax %Eop %Cmax 2 47.65 48.11 1.53 0.17 4 53.24 52.88 0.87 0.54 6 57.02 58.75 0.62 0.47 Figure24showsCmax consideringworkinghours.Wesee that the resultsof thealgorithmare significantly improvedcomparedwith industry. For twomachinesperstage, thedifference isalmosta halvingofCmax from186to96.52h. In thecaseof fourmachines, it remainsbelowhalf (93and43.82h). In thecaseofsixmachines, theresultsarebetterbyalmost three timeswithregardtoCmax (62hversus 25.57h). Figure25showsEop accordingto theprocessingtimeofmachines (Table3).Wesee that theB&B algorithmsignificantly improvesupontheresultsobtainedin industry. For twomachinesperstage, theEop objective is reduced toalmosthalf, and for four andsixmachines, theEop objectiveof our algorithmisreducedto less thanhalf. Table20showsthepercentageofdegradationof theresultsof industryandthoseselectedfrom thePareto frontobtainedfromthebi-objectiveGAcomparedwithresultsof theB&Balgorithm. Weobserve that thedegradationofEop andCmax observed in industryare closer toorhigher than50%.ComparingB&Bwithourbi-objectiveGA,weobserve thatourresultsare less than1.53% worse forEop and0.54%forCmax.Dueto thefact that theB&Balgorithmfindstheglobaloptimum, wedemonstrate thequalityofouralgorithm. 8.Conclusions Ourmaincontributionsaremultifold: (1) Weformulatedthecomplexproblemof thereal-life industryenvironmentof tortillaproduction consideringtwooptimizationcriteria: total completiontimeandenergyconsumption; 93